A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.

1, 2, 0, 0, 2, 2, 0, 2, 2, 4, 0, 2, 4, 6, 4, 0, 2, 2, 12, 12, 4, 0, 2, 6, 30, 18, 8, 0, 0, 2, 2, 44, 44, 32, 4, 0, 0, 2, 6, 82, 76, 74, 16, 0, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0

Offset

0,2

Comments

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

Except for the k = 0 column and the n = 0 and n = 1 rows, this is the triangle appearing on page 3 of Lenormand, which is A319411. Unlike A318928, we do not here require that a(n) >= 1.

The n = 0 row is chosen to ensure that the row-sums are A000079, although the empty word arguably has indeterminate runs-resistance.

Links

Table of n, a(n) for n=0..77.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Example

Triangle begins:
   1
   2   0
   0   2   2
   0   2   2   4
   0   2   4   6   4
   0   2   2  12  12   4
   0   2   6  30  18   8   0
   0   2   2  44  44  32   4   0
   0   2   6  82  76  74  16   0   0
   0   2   4 144 138 172  52   0   0   0
   0   2   6 258 248 350 156   4   0   0   0
   0   2   2 426 452 734 404  28   0   0   0   0
For example, row n = 4 counts the following words:
  0000  0011  0001  0010
  1111  0101  0110  0100
        1010  0111  1011
        1100  1000  1101
              1001
              1110

Mathematica

runsres[q_]:=If[Length[q]==1, 0, Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1];
Table[Length[Select[Tuples[{0, 1}, n], runsres[#]==k&]], {n, 0, 10}, {k, 0, n}]

Crossrefs

Row sums are A000079.

Column k = 2 is A319410.

Column k = 3 is 2 * A329745.

The version for compositions is A329744.

The version for partitions is A329746.

The number of nonzero entries in row n > 0 is A319412(n).

The runs-resistance of the binary expansion of n is A318928.

Cf. A001037, A096365, A225485, A245563, A319411, A325280, A329747, A329750.

Sequence in context: A029361 A275966 A284059 * A356018 A107502 A230419

Adjacent sequences: A329764 A329765 A329766 * A329768 A329769 A329770

Keyword

nonn,tabl

Author

Gus Wiseman, Nov 21 2019

Status

approved